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University of Groningen
Analytical solution for facilitated transport across a membrane
Marzouqi, Mohamed Hassan Al-; Hogendoorn, Kees J.A.; Versteeg, Geert
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Chemical Engineering Science
DOI:
10.1016/S0009-2509(02)00285-3
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Marzouqi, M. H. A., Hogendoorn, K. J. A., & Versteeg, G. F. (2002). Analytical solution for facilitated
transport across a membrane. Chemical Engineering Science, 57(22), 4817-4817. DOI: 10.1016/S00092509(02)00285-3
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Chemical Engineering Science 57 (2002) 4817 – 4829
www.elsevier.com/locate/ces
Analytical solution for facilitated transport across a membrane
Mohamed Hassan Al-Marzouqia;∗ , Kees J. A. Hogendoornb , Geert F. Versteegb
a Chemical
and Petroleum Engineering Department, UAE University, P.O. Box 17555, Al-Ain, United Arab Emirates
of Chemical Technology, University of Twente, P.O. Box 217 7500 AE Enschede, The Netherlands
b Faculty
Abstract
An analytical expression for the facilitation factor of component A across a liquid membrane is derived in case of an instantaneous
reaction A(g) + B(l) ⇔ AB(l) inside the liquid membrane. The present expression has been derived based on the analytical results of
Olander (A.I.Ch.E. J. 6(2) (1960) 233) obtained for the enhancement factor for G–L systems with bulk. The analytical expression for
the facilitation factor allows for arbitrary di;usivities of all species involved and does not contain any simpli<cation or approximations.
The facilitation factor starts from the value of unity, goes through a maximum and then reduces back to unity as the equilibrium constant
is increased. The maximum facilitation factor occurs at higher values of the equilibrium constant as the ratio of the permeate-complex
over carrier di;usivity is reduced whereas the maximum facilitation factor occurs at the same value of the equilibrium constant for
all values of DA =DB (ratio of the permeate over carrier di;usivity). A similar behavior is seen for the =ux of A as a function of the
equilibrium constant. The facilitation factor remains constant with changes in the <lm thickness whereas the =ux of A reduces with an
increase in the thickness of the <lm. A linear increase of the facilitation factor and =ux of A are seen with increasing initial carrier
concentration.
? 2002 Elsevier Science Ltd. All rights reserved.
Keywords: Facilitated transport; Di;usion coe@cients; Liquid membranes; Instantaneous reaction; Enhancement factor
1. Introduction
Facilitated transport is a process in which a chemical carrier binds reversibly and selectively with permeate species
at a feed side of a <lm, transports the permeate through the
<lm and, then, releases it at the permeate side of the <lm.
In this process, chemical reaction and di;usion occur simultaneously in the system which accelerates the transport of
the permeate species through the <lm. The <lm may either
be an immobilized liquid <lm (e.g. a liquid impregnated
in a porous inert structure) in which all species can move
freely and the porous structure is an inert or a membrane in
which the porous structure has an active contribution in the
facilitated transport (e.g. ion selective membranes). There
are several articles that describe facilitated transport in detail (Ward, 1970; Schultz, Goddard, & Suchdeo, 1974a,b;
∗
Corresponding author.
E-mail addresses: [email protected] (M. H. Al-Marzouqi),
[email protected] (J. A Hogendoorn),
[email protected] (G. F. Versteeg).
Smith, Lander, & Quinn, 1977; Goddard, 1977; Way,
Noble, Flynn, & Sloan, 1982).
Facilitated transport has received a great deal of attention because of its numerous promising applications.
Examples of these applications are transport of O2 through
hemoglobin solutions (Scholander 1960) CO through <lms
of cuprous chloride solutions (Smith and Quinn, 1980),
CO2 through liquid membranes for various amine solutions
(Teramoto et al., 1997) CO2 in ion exchange membranes
for di;erent ionomer <lms (Noble, Pellegrino, Grosgogeat,
Sperry, & Way, 1988; Yamaguchi, Koval, & Noble, 1996),
and CO2 through amine/polymer solution (Yamaguchi,
Boetje, Koval, Noble, & Bowman, 1995). Separation and
transport of a CO2 =N2 mixture through a liquid membrane
of diethanolamine (Guha, Majumdar, & Sirkar, 1990),
CO2 =CH4 through a liquid membrane of amine solutions
(Teramoto, Nakai, & Ohnishi, 1996) and removal of CH4
from a ternary mixture of CO2 , H2 S and CH4 (Way &
Noble, 1989), and H2 S from CO2 (Kreulen, Smolders, &
Versteeg, 1993) are other examples of the possible applications. Some other applications include transport and
separation of 1-hexane and 1,5-hexadiene through a <lm of
0009-2509/02/$ - see front matter ? 2002 Elsevier Science Ltd. All rights reserved.
PII: S 0 0 0 9 - 2 5 0 9 ( 0 2 ) 0 0 2 8 5 - 3
4818
M. H. Al-Marzouqi et al. / Chemical Engineering Science 57 (2002) 4817–4829
a Na<on membrane (Koval & Spontarellit, 1988), ethene
and ethane through Na<on membranes (Eriksen, Aksnes, &
Dahl, 1993), ole<n/para@n mixture through polymer membranes (Ho & Dalrymple, 1994; VanZyl & Linkov, 1997),
hydrocarbons mixture using ion exchange membranes
(VanZyl, Kerres, & Cui, 1997) and styrene and ethylbenzene through ionomer membranes (Koval, Spontarelli, &
Noble, 1989; Koval, Spontarelli, & Thoen, 1992).
2. Literature review for the reaction A(g) + B(l) ⇔ AB(l)
The most common generalized and overall reaction
scheme for the transport of a gaseous component across a
liquid <lm reported in the literature is
A(g) + B(l) ⇔ AB(l);
(1)
where A, B and AB are permeate species, mobile carrier, and
permeate–carrier complex, respectively. Since the governing
equations that describe the steady state transport of permeate species across the liquid <lm of simultaneous chemical
reaction and di;usion of the above reaction are coupled and
usually nonlinear, a generally applicable analytical solution
for arbitrary kinetics is not possible.
There have been several attempts to obtain analytical
solutions for such systems, however, most of the analytical
approaches reported are only valid for simpli<ed situations
not completely representing the actual process in the membrane. An analytical solution for the steady state facilitation
factor, which is de<ned as the ratio of the actual solute =ux
(with chemical reaction) to the solute di;usion =ux only,
was developed by Smith and Quinn (1979) in rectangular
coordinates assuming equilibrium concentrations. The authors linearized the reaction rate expression by assuming
that there is a large excess of mobile carrier. Smith and
Quinn additionally assumed equal di;usivity of carrier and
complex in their analysis. They reported the correct behavior for asymptotic reaction- and di;usion-limited cases at
low equilibrium constant. However, the accuracy of the facilitation factor obtained by this approach decreases at large
values of the equilibrium constant (refer to Fig. 1). Noble,
Way, and Power (1986) derived an expression for the facilitation factor using the similar approach as Smith and Quinn
which also accounts for external mass transfer resistance.
The equation for the facilitation factor obtained by the authors reduces to Smith and Quinn’s for the case of no external mass transfer. Also in this paper, the authors used equal
di;usivity of the carrier and complex in the analysis. The
accuracy of the facilitation factor obtained by the authors
decreases at high equilibrium constant, similar to Smith and
Quinn. An improved method for the evaluation of facilitation factor is reported by Jemaa and Noble (1992). The
authors empirically determined a nonzero permeate concentration at the exit of the membrane by matching the facilitation factor resulting from Smith and Quinn’s model with the
one computed numerically by Kemena, Noble, and Kemp
Fig. 1. Comparison of the e;ect of facilitation factor as a function
of equilibrium constant between Smith and Quinn and the exact solution for DAB = DA = 10−9 (m2 =s), = 1 (−), CA0 = 100 (mol=m3 ),
CAL = 1 (mol=m3 ), CBinitial = 1000 (mol=m3 ), and L = 0:001 (m).
(1983). Again, in this improved method equal di;usivities
of the carrier and complex were assumed. However, the introduction of an empirical <t parameter is not very elegant.
Noble (1990) derived a model for the facilitation of neutral
molecules such as O2 across a <xed site carrier membrane.
The analysis of the analytical determination of the facilitation factor is similar to the one presented by Smith and
Quinn (1979) for equal carrier and complex di;usivities.
Basaran, Burban, and Auvil (1989) presented two models for the facilitated transport of unequal carrier and
permeate-carrier complex di;usivities which also permit
arbitrary kinetic rates. The <rst method was derived for
very low Damkohler numbers (kr L2 =DA ) and solved analytically using regular perturbation analysis. This has limited
industrial application because for low Damkohler number,
the contribution of the carrier to facilitated transport is
low. The second model was solved numerically using a
Galerkin/<nite element method that was solved by an iteration technique using Newton’s method. Teramoto (1994)
presented an approximate solution for the facilitation factor.
Using trial & error calculations, approximate facilitation
factors were calculated and compared with several reported
numerical solutions with a good agreement. The model is
valid for both equal and unequal di;usivities.
A model by Chaara and Noble (1989) describes di;usion, chemical reaction and convection across a liquid <lm.
The authors obtained a facilitation factor for equal di;usivities of carrier and a complex by assuming equilibrium composition and a constant carrier concentration. This model
is an extension of the expression obtained by Noble et al.
(1986) accounting for the convective =ow across the <lm,
however, with the same limitation(s).
Beside these semi-analytical solutions, of course, several authors solved the governing equations that describe
the steady state transport of permeate species across the
liquid <lm of the above system numerically. Jain and
Schultz (1982) solved the system of di;erential equations
M. H. Al-Marzouqi et al. / Chemical Engineering Science 57 (2002) 4817–4829
using orthogonal collocation for both equal and unequal
di;usivities. Kemena et al. (1983) studied the optimization
of the facilitation factor for equal di;usivities of carrier
and complex using a numerical technique. Kirkkoprudindi
and Noble (1989) presented the numerical results obtained
for multiple site transport using equal carrier and complex
di;usivities. Dindi, Noble, and Koval (1992) solved the
model for the competitive facilitated transport numerically
for unequal di;usivities of carrier and complex. An equilibrium composition with a constant concentration of carrier
was used. The equation obtained for the facilitation factor in
this study reduced to the one of Smith and Quinn (1979) for
one permeate and equal di;usivities of carrier and complex.
Unsteady state competitive facilitated transport of two
gases through the membrane was determined numerically
by Niiya and Noble (1985). This model is compared with
other models for the case of steady state and “equilibrium
core” with good agreement. Folkner and Noble (1983) reported the transient =ux for one-dimensional transport in
rectangular, cylindrical and spherical co-ordinates using
numerical techniques.
Nearly all models mentioned in the previous section are
either inaccurate (especially the approximate analytical solutions as the problem is oversimpli<ed) or elaborate to
work with (numerical models). Except for the solutions
by Basaran et al. (1989) (partly numerical) and Teramoto
(1994) (using an assumption on the concentration of B),
all other analytical solutions have not been able to accurately predict the facilitation factor over the entire range
from di;usion- to reaction-limited mass transport. Furthermore, all analytical analyses have been restricted to equal
di;usivity of the mobile carrier and permeate-carrier complex except the analyses presented by Basaran et al. and
Teramoto. It is quite clear that there is a need for a simple
method to reliably predict the facilitation factor.
The aim of this paper is to derive an analytical equation
for the facilitation factor that allows for unequal di;usivity
of the carrier and permeate-carrier complex for reactions
instantaneous with respect to mass transfer. The method
is based on a simple equation derived by Olander (1960).
Originally Olander derived equations for instantaneous reactions for systems with a liquid bulk, but the solution is
also applicable to instantaneous reactions in liquid <lms.
Previously, Kreulen et al. (1993) showed that the facilitation factor for the reaction A + B ↔ C + D using equal
di;usivities of all components could be calculated analytically using Olander’s approach. It will be shown that the
present method, based on Olander’s equations, is applicable for instantaneous reactions with arbitrary values of the
di;usivities of the various species.
3. Theory
The governing equations for steady state, one-dimensional
transport in a rectangular geometry of the overall reaction
scheme shown in Eq. (1) are as follows:
d 2 CA
CAB
DA
= 0;
+ k1 −CA CB +
d x2
K
d 2 CB
CAB
DB
= 0;
−C
+
k
C
+
1
A B
d x2
K
d 2 CAB
CAB
= 0;
DAB
C
+
k
C
−
1
A B
d x2
K
4819
(2)
(3)
(4)
where CA , CB , CAB are the concentration of permeate (A),
mobile carrier (B), and permeate–carrier complex (AB), respectively, DA , DB , and DAB are the di;usivity of A, B and
AB, respectively, k1 is the forward reaction rate, K is the
equilibrium constant, and x is the distance.
The boundary conditions for Eqs. (2)–(4) are given as
x = 0;
CA = CA0 ;
dCAB
dCB
=
= 0;
dx
dx
(5)
x = L;
CA = CAL ;
dCAB
dCB
=
= 0:
dx
dx
(6)
Eqs. (2)–(4) can be simpli<ed to
DA
d 2 CA
d 2 CAB
+
D
= 0;
AB
d x2
d x2
(7)
DB
d 2 CB
d 2 CAB
+
D
= 0:
AB
d x2
d x2
(8)
For an instantaneous reaction
K=
CAB
:
CA CB
(9)
The boundary conditions for an instantaneous reaction
change to
x = 0;
CA = CA0 ;
DB
dCB
dCAB
+ DAB
= 0;
dx
dx
CAB0 = KCA0 CB0 ;
x = L;
CA = CAL ;
CABL = KCAL CBL :
(10)
DB
dCB
dCAB
+ DAB
= 0;
dx
dx
(11)
The problem is now identical to the problem of mass transfer with simultaneous reactions according to the <lm model
for systems with a bulk. This model has been successfully
solved by Olander (1960) not only for the present instantaneous reaction of A(g)+B(l) ⇔ AB(l) but also for other instantaneous reactions including A(g) + B(l) ⇔ C(l) + D(l).
If the solution of Olander for the present reaction is considered, one can see that the concentration of B at x = L is
required. For a system with bulk the concentration of B is
imposed by the liquid loading and the assumption of equilibrium in the liquid bulk. However, for the present case the
4820
M. H. Al-Marzouqi et al. / Chemical Engineering Science 57 (2002) 4817–4829
following condition has to be satis<ed:
L
(CB (x) + CAB (x)) d x = CBinitial L:
0
(12)
It can be shown however that (refer to Appendix A):
DAB
C∗
CAB =
:
(13)
CB +
DB
DB
This gives for the concentration of B at x = L:
1 + KCA0
;
CBL = CB0
1 + KCAL
(14)
where CB0 can be obtained from the equation below:
C∗
CB0 =
(15)
DB (1 + KCA0 )
or
C∗
:
(16)
CBL =
DB (1 + KCAL )
Fig. 2. E;ect of equilibrium constant on facilitation factor for di;erent
values of DA =DB for the case of DB = DAB = 10−9 (m2 =s); L = 0:001 (m);
CA0 = 10 (mol=m3 ); CAL = 1 (mol=m3 ); CBinitial = 1000 (mol=m3 ).
In this equation C ∗ is de<ned according to implicit equation
(B.28) or the identical expression (B.44) in Appendix B.
The facilitation factor can now be calculated to be
KC ∗
F =1+
DA (1 + (DAB =DB )KCAL )(1 + (DAB =DB )KCA0 )
(17)
and the =ux according to
DA
(18)
JA = F
(CA0 − CAL ):
L
Eq. (17) starts at the value of 1, increases to its maximum
value, and, then, reduces to the value of unity as the equilibrium constant is increased from a low to high value.
4. Results and discussion
Fig. 2 shows a typical relationship between the facilitation factor and equilibrium constant for di;erent values of
DA =DB , which is the ratio of the di;usivity of permeate to
that of carrier. As expected, the facilitation factor starts at
the value of unity, increases to its maximum value, and then
drops back to unity as the value of the equilibrium constant
is increased. As is presented in Fig. 2, the maximum occurs
at the same value of the equilibrium constant for all values
of DA =DB . However, the maximum value of the facilitation
factor increases as the ratio of DA =DB reduces. The relationship between the facilitation factor and equilibrium constant
for di;erent values of ( = DAB =DB ) is presented in Fig.
3a. Similar to Fig. 2, the facilitation factor starts at the value
of 1, increases to its maximum value, and then drops back to
unity as the value of the equilibrium constant is increased.
The maximum value of the facilitation factor increases as
the ratio of reduces, however, the maximum facilitation
factor occurs at a higher equilibrium constant as the value
of is reduced. The relationship between the =ux of A
Fig. 3. (a) E;ect of equilibrium constant on facilitation factor for di;erent
values of (DAB =DB ) for the case DAB =DA =10−9 (m2 =s); L = 0:001 (m),
CA0 = 10 (mol=m3 ); CAL = 1 (mol=m3 ), CBinitial = 1000 (mol=m3 ). (b) Effect of equilibrium constant on =ux of A for di;erent values of (DAB =DB )
for the case DAB =DA =10−9 (m2 =s), L=0:001 (m), CA0 =10 (mol=m3 ),
CAL = 1 (mol=m3 ), CBinitial = 1000 (mol=m3 ).
and equilibrium constant for di;erent values of ( =
DAB =DB ) is presented in Fig. 3b, which shows a similar
behavior as Fig. 3a. Fig. 4a shows the relationship between
the facilitation factor and equilibrium constant for di;erent values of the initial permeate concentration (CA0 ). As
expected, the facilitation factor goes through a maximum
for all values of CA0 , however, the maximum value of the
facilitation factor increases as CA0 (or CA0 =CAL ) is reduced.
In addition, this maximum occurs at higher values of the
equilibrium constant as CA0 is reduced. The relationship
between the =ux of A and equilibrium constant for di;erent
M. H. Al-Marzouqi et al. / Chemical Engineering Science 57 (2002) 4817–4829
4821
Fig. 6. Facilitation factor and =ux of A as a function of initial carrier
concentration for di;erent values of equilibrium constant for the case
of DAB = DA = 10−9 (m2 =s), DB = 5 × 10−10 (m2 =s), L = 0:001 (m),
CA0 = 10 (mol=m3 ), CAL = 1 (mol=m3 ), CBinitial = 1000 (mol=m3 ).
Fig. 4. (a) E;ect of equilibrium constant on facilitation factor for
di;erent values of initial permeate concentration for the case of
DAB = DA = DB = 10−9 (m2 =s), L = 0:001 (m), CAL = 1 (mol=m3 ),
CBinitial = 1000 (mol=m3 ). (b) E;ect of equilibrium constant on =ux
of A for di;erent values of initial permeate concentration for the case
of DAB = DA = DB = 10−9 (m2 =s), L = 0:001 (m), CAL = 1 (mol=m3 ),
CBinitial = 1000 (mol=m3 ).
Fig. 5. Facilitation factor and =ux of A as a function of <lm
thickness for the case of DAB = DA = 10−9 (m2 =s), DB = 5 ×
10−10 (m2 =s), K = 1 (m3 =mol), CA0 = 10 (mol=m3 ), CAL = 1 (mol=m3 ),
CBinitial = 1000 (mol=m3 ).
Fig. 7. Permeate concentration pro<le for di;erent values of equilibrium constants for the case of DAB = DB = 10−9 (m2 =s),
DA = 5 × 10−10 (m2 =s), L = 0:001 (m), CA0 = 250 (mol=m3 ),
CAL = 1 (mol=m3 ), CBinitial = 1000 (mol=m3 ); F corresponds to facilitation factor. (b) Carrier and complex concentration pro<le for di;erent
values of equilibrium constants for the case of DAB = DB = 10−9 (m2 =s),
Da = 5 × 10−10 (m2 =s), L = 0:001 (m), CA0 = 250 (mol=m3 ),
CAL = 1 (mol=m3 ), CBinitial = 1000 (mol=m3 ), F corresponds to facilitation factor.
values of initial permeate concentration (CA0 ) is presented
in Fig. 4b. The =ux of A goes through a maximum for all
values of CA0 , however, the maximum value of the =ux of
A increases as CA0 (or CA0 =CAL ) is increased. In addition,
this maximum occurs at higher values of the equilibrium
constant as CA0 is reduced. Fig. 5 shows the relationship
between the facilitation factor and =ux of A as a function of
membrane thickness. The facilitation factor remains constant as the membrane thickness is increased, whereas, the
=ux of A drops with the membrane thickness linearly. As
presented in the last part of Appendix B, C ∗ is independent
of the thickness, causing the facilitation factor to remain
unchanged with the thickness of the <lm. The relationship
between the facilitation factor and =ux of A as a function
of the initial carrier concentration for di;erent values of the
equilibrium constant is presented in Fig. 6. Both the facilitation factor and =ux of A increase linearly with increasing
initial carrier concentration. Fig. 7a shows the permeate
4822
M. H. Al-Marzouqi et al. / Chemical Engineering Science 57 (2002) 4817–4829
Fig. 8. E;ect of equilibrium constant on C ∗ =Db for di;erent values of
(DAB =DB ) for the case of DAB = DA = 10−9 (m2 =s), L = 0:001 (m),
CA0 = 10 (mol=m3 ), CAL = 1 (mol=m3 ), CBinitial = 1000 (mol=m3 ).
concentration pro<le for di;erent values of the equilibrium
constant. The permeate concentration is at its maximum
at the high-pressure side of the membrane and then drops
across the membrane until it reaches the permeate concentration at low-pressure side of the membrane. Carrier
and complex concentration pro<les for the corresponding
values of the equilibrium constant are presented in Fig.
7b. The concentration of the carrier at the high-pressure
side of the <lm is initially low and increases along the
thickness of the <lm, whereas, the permeate–carrier concentration is high at x = 0 and then decreases to its minimum value at the low-pressure side of the <lm (x = L).
The relationship between C ∗ =DB and equilibrium constant
for di;erent values of ( = DAB =DB ) is presented in
Fig. 8. The value of C ∗ =DB starts at CBinitial for low equilibrium constants and then it goes through a S-shaped curve
and <nally converges to the value of CBinitial as the equilibrium constant is increased. It is also important to note
that, for = 1; C ∗ =DB will remain constant and it has
the same value of CBinitial for all values of the equilibrium
constant.
The present contribution deals with the reaction A(g) +
B(l) ⇔ AB(l) as it is the most frequently encountered from
of facilitated transport. However, for other reactions the appropriate solution of Olander (1960), derived for G–L systems with bulk) can also be used in a similar way to derive
the accompanying (analytical) expression for the facilitation
factor across a membrane.
5. Conclusion
An analytical equation for the facilitation factor, which
allows for unequal di;usivity of the carrier and permeate–
carrier complex for reactions instantaneous with respect to
mass transfer was derived for the system of A(g) + B(l) ⇔
AB(l). The present analytical solution has been derived
using the results of Olander (1960) for instantaneous reactions in G–L systems with bulk. The analytical solution
can be applied for any set of di;usivities of reactants
and product and does not contain any approximation or
simpli<cation.
The facilitation factor starts from the value of unity, goes
through a maximum and then reduces back to unity as the
equilibrium constant is increased with the changes in di;usion coe@cients of A, B and/or AB, and with the changes
in initial permeate concentrations. The maximum facilitation factor occurs at higher values of the equilibrium constant as the ratio of the permeate-complex over carrier diffusivity is reduced whereas the maximum facilitation factor occurs at the same value of equilibrium constant for all
values of DA =DB (ratio of the permeate over carrier di;usivity). A similar behavior is seen for the =ux of A as a
function of equilibrium constant. Facilitation factor remains
constant with changes in the <lm thickness whereas the =ux
of A reduces with an increase in the thickness of the <lm.
A linear increase on facilitation factor and =ux of A are
seen with increasing initial carrier concentration. The values of C ∗ =DB starts at CBinitial for low equilibrium constants
and then it goes through a S-shaped curve and <nally converges at the value of CBinitial as the equilibrium constant is
increased.
Notation
CA
CA0
CAL
CB
CAB
CBinitial
DA
DB
DAB
F
JA
k1
K
L
x
concentration of permeate A, mol=m3
initial concentration of permeate A, mol=m3
<nal concentration of permeate A, mol=m3
concentration of mobile carrier B, mol=m3
concentration of permeate-carrier complex AB,
mol=m3
initial concentration of carrier, mol=m3
di;usion coe@cient of permeate A, m2 =s
di;usion coe@cient of mobile carrier B, m2 =s
di;usion coe@cient of permeate-carrier complex AB, m2 =s
facilitation factor, dimensionless
=ux of A across membrane, mol=m2 s
forward reaction rate constant, m3 =mol s
equilibrium constant, m3 =mol
characteristic length or thickness of membrane,
m
position, m
ratio of the carrier and complex di;usivities,
dimensionless
Acknowledgements
The authors would like to thank United Arab Emirates
University for the support. The support of the Chemical
Technology department of the University of Twente is
highly appreciated.
M. H. Al-Marzouqi et al. / Chemical Engineering Science 57 (2002) 4817–4829
4823
Appendix A. Derivation of expression for the facilitation
factor
Di;erentiating Eq. (A.12) and applying Eq. (A.9b) results
in a3 to be 0. Therefore, Eq. (A.12) reduces to
The governing equations are:
d 2 CA
CAB
DA
= 0;
+ k1 −CA CB +
d x2
K
d 2 CB
CAB
= 0;
−C
DB
+
k
C
+
1
A B
d x2
K
d 2 CAB
CAB
= 0:
DAB
C
+
k
C
−
1
A
B
d x2
K
DB CB + DAB CAB = a4
(A.1)
(A.2)
(A.3)
The boundary conditions for Eqs. (A.1)–(A.3) are given as
x = 0;
x = L;
CA = CA0 ;
dCB
dCAB
=
= 0;
dx
dx
(A.4)
CA = CAL ;
dCAB
dCB
=
= 0:
dx
dx
(A.5)
DA
d 2 CA
d 2 CAB
+
D
= 0;
AB
d x2
d x2
2
(A.6)
DB CB + DAB KCA CB = a4 :
Solving for CB :
a4
CB =
;
DB (1 + KCA )
d CB
d CAB
+ DAB
= 0:
DB
d x2
d x2
(A.7)
For instantaneous reaction
=
(A.8)
a1 =
From Eqs. (A.11) and (A.8):
x = 0;
CA = CA0 ;
(A.9a)
x = L;
CA = CAL :
(A.9b)
dCB
dCAB
DB
+ DAB
= 0;
dx
dx
D A CA + KCA
a4 = a1 x + a 2 ;
(1 + KCA )
Solutions to Eqs. (A.6) and (A.7) are:
DA CA + DAB CAB = a1 x + a2 ;
(A.11)
DB CB + DAB CAB = a3 x + a4 ;
(A.12)
where a1 ; a2 ; a3 and a4 are the constants that will be determined.
(A.17)
where a1 ; a2 and a4 need to be determined.
Using boundary condition (A.9a), Eq. (A.17) results in:
KCA0
a4 = a2
(1 + KCA0 )
(A.18)
and using boundary condition (A.10), Eq. (A.17) results in:
KCAL
a4 = a 1 L + a 2 :
(1 + KCAL )
(A.19)
Combining Eqs. (A.18) and (A.19), and solving for a1 :
(A.20)
To determine a4 : Eq. (A.15) is evaluated at x = 0 and x = L:
At x = 0;
CB0 =
a4
;
DB (1 + KCA0 )
(A.21)
At x = L;
CBL =
(A.10)
(A.16)
Combination of Eqs. (A.16) and (A.15) leads to
DA (CAL − CA0 ) + Ka4 (CAL =(1 + KCAL ) − CA0 =(1 + KCA0 ))
:
L
Eqs. (A.6)–(A.8) can be solved analytically using the appropriate boundary conditions. The boundary conditions for
an instantaneous reaction are:
(A.15)
DAB
:
DB
DA CAL + CAB
K=
:
CA CB
(A.14)
where
DA CA0 + 2
x = 0 & x = L;
or
DA CA + DAB KCA CB = a1 x + a2 :
This system is simpli<ed by addition of Eqs. (A.1) and (A.3)
and (A.2) and (A.3), the resultant equations are:
(A.13)
a4
;
DB (1 + KCAL )
(A.22)
where CB0 and CBL need to be determined. Solving for a4
and equating Eqs. (A.21) and (A.22) leads to
CBL = CB0
1 + KCA0
:
1 + KCAL
(A.23)
Therefore
CBL + CABL = CBL + KCAL CBL
= CBL (1 + KCAL ):
(A.24)
4824
M. H. Al-Marzouqi et al. / Chemical Engineering Science 57 (2002) 4817–4829
Combining Eqs. (A.23) with (A.24):
CBL + CABL = CB0
From Eq. (A.8):
1 + KCA0
(1 + KCAL )
1 + KCAL
CAB = KCA CB :
(A.33)
The facilitation factor can now be calculated to be
= CB0 (1 + KCA0 )
−a1
:
−DA ((CAL − CA0 )=L)
(A.34)
= CB0 + KCA0 CB0
F=
= CB0 + CAB0 :
Combination of Eqs. (A.27) and (A.34) and simpli<cation
leads to
Or
a4
C∗
CB + CAB = constant =
=
:
DB
DB
(A.25)
F =1+
KC ∗
:
DA (1 + (DAB =DB )KCAL )(1 + (DAB =DB )KCA0 )
(A.35)
From Eq. (A.25):
a4 = C ∗ :
(A.26)
Eq. (A.20) can be simpli<ed to
a1 =
Eq. (A.35) starts at the value of unity, increases to its maximum value, and then decreases to unity as K is increased.
The expression for C ∗ is derived in Appendix B.
DA (CAL − CA0 ) + KC ∗ [(CAL − CA0 )=((1 + KCAL )(1 + KCA0 ))]
L
To calculate the concentration gradient of A; B and AB the
following steps are taken:
From Eqs. (A.17) and (A.26):
KCA
D A CA + C ∗ = a1 x + a 2 ;
(1 + KCA )
(A.28)
where a1 is given by Eq. (A.27) and a2 can be evaluated by
combining Eqs. (A.18) and (A.26):
KCA0
C ∗:
a2 = DA CA0 + (1 + KCA0 )
(A.29)
Eq. (A.28) with the constants a1 and a2 (Eqs. (A.27) and
(A.29)) is a quadratic equation, which can be solved for the
concentration of A(CA ) in the <lm.
√
−b + b2 − 4ac
CA =
;
(A.30)
2a
where
b = DA + KC ∗ − Ka1 x − a2 K;
c = −(a1 x + a2 ):
The concentrations of B(CB ) and AB(CAB ) in the <lm can
be obtained taking the following steps:
From Eqs. (A.22) and (A.26):
(A.31)
Note that Eq. (A.22) is valid for any point in the <lm and
CAB can be calculated using the following equations:
From Eq. (A.25):
CAB =
(C ∗ =DB ) − CB
:
Appendix B. Derivation of expression for C ∗
Use
0
L
(CB + CAB ) d x = CBinitial L:
(B.1)
We know (from Eq. (A.15))
DB CB + DAB CAB = C ∗ :
(B.2)
Solving for CAB :
CAB =
C ∗ − D B CB
C∗
CB
=
−
DAB
DAB
where =
Substituting Eq. (B.3) into Eq. (B.1):
L
C∗
CB
d x = CBinitial L:
CB +
−
DAB
0
a = KDA ;
C∗
CB =
:
DB (1 + KCA )
(A.27)
(A.32)
Integrating Eq. (B.4):
L
C∗
1
L+ 1−
CB d x = CBinitial L:
DAB
0
DAB
:
DB
(B.3)
(B.4)
(B.5)
Now, lets work with
L
CB d x:
0
From Eq. (B.2) and the equilibrium concentration (CAB =
KCA CB ):
CB + KCA CB =
C∗
;
DB
(B.6)
M. H. Al-Marzouqi et al. / Chemical Engineering Science 57 (2002) 4817–4829
or
A3 = K;
C∗
:
DB (1 + KCA )
CB =
Thus
L
CB d x =
(B.7)
L
0
=
A6 = A3 B = KB = A2 ;
L
(B.8)
dx
1 + KCA
∗ L
C
DB
A7 =
A4
4aa1 − 2AB 2(DA − KC ∗ + Ka2 )
=
=
= 2A9 ;
B2
B2
Ka1
A8 =
A5
4aa2 + A2
=
B2
B2
=
dx
;
1 + K(−b + b2 − 4ac)=2a
0
A4 = 4aa1 − 2AB;
A5 = 4aa2 + A2 ;
C∗
dx
0 DB (1 + KCA )
C∗ L
dx
=
;
DB 0 1 + KCA
0
C∗
DB
4825
(B.9)
A9 =
where CA , a; b and c are given as (refer to Appendix A)
√
−b + b2 − 4ac
;
(A.30)
CA =
2a
a = KDA ;
A = DA + KC ∗ − a2 K
and
B = Ka1 ;
2KDA − K(DA + KC ∗ − Ka2 )
(K)2 a1
=
DA − KC ∗ + Ka2
;
Ka1
Eq. (B.9) can be written as
C∗ L
2a d x
DB 0 2a + K[ − A + Bx + (A − Bx)2 + 4aa1 x + 4aa2 ]
2aC ∗
=
DB
2aC ∗
=
DB
=
2aC ∗
DB
=
2aC ∗
DB
= A10
0
L
L
0
2a − KA + KBx + K[
0
0
A1 + A2 x + A3 B[
L
L
0
dx
A2
− 2ABx + B2 x2 + 4aa1 x + 4aa2 ]
A2 (x + (A1 =A2 )) + A6 [
L
dx
A2 (x + A9 ) + A6 [
(B.14)
(B.15)
x 2 + A7 x + A 8 ]
x 2 + A7 x + A 8 ]
(B.11)
(B.13)
+ (A4 =B2 )x + (A5 =B2 )]
dx
(B.17)
(B.12)
dx
x2
dx
A2 [(x + A9 ) + x2 + 2A9 x + A8 ]
(B.10)
dx
A1 + A 2 x + A 6 [ x 2 + A 7 x + A 8 ]
L
0
= A10
dx
A1 + A 2 x + A 3 [ B 2 x 2 + A 4 x + A 5 ]
L
0
2aC ∗
2KDA C ∗
=
:
DB
DB
Continuing with Eq. (B.16)):
c = −(a1 x + a2 ) = −a1 x − a2 :
2aC ∗
=
DB
A1
2a − KA
=
A2
KB
=
A10 =
b = DA + KC ∗ − Ka1 x − a2 K = A − Bx;
4KDA a2 + (DA + KC ∗ − a2 K)2
;
(Ka1 )2
;
(B.16)
where
A1 = 2a − KA;
2
A2 = KB = (K) a1 ;
=
A10
A2
0
L
(x + A9 ) +
dx
:
x2 + 2A9 x + A8
(B.18)
4826
M. H. Al-Marzouqi et al. / Chemical Engineering Science 57 (2002) 4817–4829
Multiplying both the numerator and denominator by the
conjugate:
A10
A2
L
0
(x + A9 ) − x2 + 2A9 x + A8 d x
[(x + A9 ) + x2 + 2A9 x + A8 ][(x + A9 ) − x2 + 2A9 x + A8 ]
A10
=
A2
A10
=
A2
=
L
0
L
(x + A9 ) − x2 + 2A9 x + A8 d x
(x + A9 )2 − (x2 + 2A9 x + A8 )
(x + A9 ) −
0
A10
A2 (A29 − A8 )
0
L
(B.20)
x2 + 2A9 x + A8 d x
− A8
(B.21)
A29
[A9 + x −
x2 + 2A9 x + A8 ] d x
(B.22)
L
L2
A10
2
(x + 2A9 x + A8 ) d x
=
A9 L +
−
2
A2 (A29 − A8 )
0
=
(B.23)
L2
A10
A9 L +
2
2
A2 (A9 − A8)
−
=
(B.19)
0
L
(x2
+ 2A9 x +
A29 )
−
(A29
− A8 ) d x
(B.24)
L
L2
A10
2 − (A2 − A ) d x
A
L
+
(x
+
A
)
−
9
9
8
9
2
A2 (A29 − A8 )
0
L2
x + A9
A
L
+
(x + A9 )2 − (A29 − A8 )
−
 9
2
2


2

A10
 − A9 − A8 ln |x + A9
=
A2 (A29 − A8 ) 
2


L

+ (x + A9 )2 − (A29 − A8 )|

(B.25)










(B.26)
0

L2
L + A9
2 − (A2 − A )
A
L
+
(L
+
A
)
−
9
8
9

 9
2
2



 


L
+
A
9


A10
A29 − A8 
  ;
=
ln
−

2
A2 (A29 − A8 ) 

+ (L + A9 )2 − (A29 − A8 ) 




2


A9
A − A8
+
ln |A9 + A8· |
A8· − 9
2
2

(B.27)
M. H. Al-Marzouqi et al. / Chemical Engineering Science 57 (2002) 4817–4829
4827
With Eq. (B.5), the following expression can now be
obtained:
C∗
1
L+ 1−
DAB

L + A9
L2
2 − (A2 − A )
A
−
L
+
(L
+
A
)
9
9
8
9



2
2





 



L + A9



A10
A29 − A8 



ln ×
−

 = CBinitial L:
2
2
 A2 (A9 − A8 ) 
+ (L + A9 )2 − (A29 − A8 ) 









A9
A29 − A8
+
A8· −
ln |A9 + A8· |
2
2


After further simpli<cation:
A29 − A8 = −
a2 = DA CA0 + ∗
4DA C
;
Ka21
(B.29)
A10
1 a1
;
=−
2
2 DB
A2 (A9 − A8 )
(B.30)
A5
4aa2 + A2
A8 = 2 =
B
B2
=
A9 =
=
4KDA a2 + (DA + KC ∗ − a2 K)2
;
(Ka1 )2
A8 =
4KDA a2 + (DA + KC ∗ − a2 K)2
(Ka1 )2
=
C5
C5
= L2 2 = L2 C6;
2
C1
a1
∗
DA − KC ∗ + Ka2
=
;
Ka1
a1 =
DA − KC ∗ + Ka2
C3
C3
= LC4;
=
=L
Ka1
a1
C1
A29 − A8 = −
2KDA − K(DA + KC − Ka2 )
(K)2 a1
(B.32)
KCA0
C ∗:
(1 + KCA0 )
(A.29)
(B.35)
(B.36)
4DA DB C ∗
C7
C7
= 2 = L2 2 = L2 C8;
C1
Ka21
a1
(B.37)
(A.27)
(L + A9 )2 − (A29 − A8 ) =
Eqs. (B.28)–(B.32) with Eqs. (A.27) and (A.29) give the
relationship between constant C ∗ and CBinitial . The equation
is nonlinear and requires an iterative solution procedure.
B.1. Dependency of C ∗ on L (refer to Fig. 5)
=
DA (CAL − CA0 ) + KC ∗ [(CAL − CA0 )=((1 + KCAL )(1 + KCA0 ))] C1
=
;
L
L
(L + L:C4)2 − L2 :C8
L2 [(1 + C4)2 − C8]
= L: [(1 + C4)2 − C8]
= L:C11;
To determine the dependency of C ∗ on the thickness of
the <lm (L), the following steps are taken:
a1 =
(B.34)
A10
C1C9 C10
1 a1
= a1 C9 =
=−
=
; (B.38)
2 DB
L
L
A2 (A29 − A8 )
DA (CAL − CA0 ) + KC ∗ [(CAL − CA0 )=((1 + KCAL )(1 + KCA0 ))]
L
a2 = DA CA0 + KCA0
C ∗ = C2;
(1 + KCA0 )
A9 =
(B.31)
A1
2a − KA
=
A2
KB
(B.28)
(B.33)
(B.39)
4828
ln
M. H. Al-Marzouqi et al. / Chemical Engineering Science 57 (2002) 4817–4829
(L + A9 +
1 1 + C4
1
C4 + −
C11
C15 = C10 1 −
2
2
C4 √
C8
+
C14 :
C6 +
2
2
(L + A9 )2 − (A29 − A8 ))
√
A 9 + A8
= ln
L + L:C4 + L:C11
√
L:C4 + L2 :C6
= ln
L(1 + C4 + C11)
C12
√
= C14:
= ln
C13
L(C4 + C6)
(B.40)
References
Rearranging Eq. (B.28):
C10
1
C∗
L2
L2 C4 +
L+ 1−
DAB
L
2
L2 C8
L + LC4
LC4 √ 2
C14
L C6 +
LC11 +
2
2
2
−
= CBinitial L:
(B.41)
After further
C∗
L+ 1−
DAB
simpli<cation, Eq. (B.41) reduces to
C10
1
1
L2 C4 +
L
2
C4 √
C8
1 + C4
C11 +
C6 +
−
C14
2
2
2
=CBinitial L
(B.42)
or
C∗
L + L:C15 = CBinitial L;
DAB
(B.43)
C∗
+ C15 = CBinitial ;
DAB
(B.44)
where
C1 = DA (CAL − CA0 )
CAL − CA0
∗
;
+ KC
(1 + KCAL )(1 + KCA0 )
C2 = DA CA0 + KCA0
C ∗ = a2 ;
(1 + KCA0 )
C3 =
DA − KC ∗ + Ka2
;
K
C4 =
C3
;
C1
C6 =
C5
;
C12
C7 = −
C8 =
C7
;
C12
C9 = −
C5 =
C10 = C1:C9;
4KDA a2 + (DA + KC ∗ − a2 K)2
;
(K)2
1
;
2DB
C11 = (1 + C4)2 − C8;
C12 = 1 + C4 + C11;
C14 = ln
C12
;
C13
4DA DB C ∗
;
K
C13 = C4 +
√
C6;
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is shown graphically in Fig. 5.
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